A Note on the Closed-Form Solution of the Multi-Layer CES Consumer Model

Sungwhee Shin, Ki Hong Choi

Research output: Contribution to journalArticlepeer-review


The advantages of the CES utility function in the consumer model are the generality of elasticity of substitution and the natural extension to multilayer CES. The disadvantage is that complex calculations are required to derive a closed-form solution of the demand functions. Deriving a closed solution of the demand function from the multi-layer CES model is almost impossible due to the exponential increase in complexity. In the consumer model, there is Shephard’s lemma, which induces the compensated demand by the’partial differentiation’ of the expenditure function. However, there is no such convenient tool for the market demand. This paper showed that market demand is induced by the ‘logarithmic partial differentiation’ of the expenditure function using the linear homogeneity of the CES function and the basic duality theory between market demand and compensated demand. This is essential for a multi-layer CES model that needs to solve both the compensated demand and the market demand simultaneously.

Original languageEnglish
Pages (from-to)65-76
Number of pages12
JournalJournal of Economic Theory and Econometrics
Issue number4
StatePublished - Dec 2021


  • Compensated Demand
  • Duality Theory
  • Household Optimization
  • Market Demand
  • Multi-Layer CES Utility
  • Self-dual


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